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SageMath
E = EllipticCurve("r1")
E.isogeny_class()
Elliptic curves in class 9680.r
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
9680.r1 | 9680w4 | \([0, 0, 0, -114587, 14925834]\) | \(22930509321/6875\) | \(49887157760000\) | \([4]\) | \(30720\) | \(1.6064\) | |
9680.r2 | 9680w3 | \([0, 0, 0, -56507, -5049814]\) | \(2749884201/73205\) | \(531198455828480\) | \([2]\) | \(30720\) | \(1.6064\) | |
9680.r3 | 9680w2 | \([0, 0, 0, -8107, 167706]\) | \(8120601/3025\) | \(21950349414400\) | \([2, 2]\) | \(15360\) | \(1.2599\) | |
9680.r4 | 9680w1 | \([0, 0, 0, 1573, 18634]\) | \(59319/55\) | \(-399097262080\) | \([2]\) | \(7680\) | \(0.91328\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 9680.r have rank \(0\).
Complex multiplication
The elliptic curves in class 9680.r do not have complex multiplication.Modular form 9680.2.a.r
sage: E.q_eigenform(10)
Isogeny matrix
sage: E.isogeny_class().matrix()
The \(i,j\) entry is the smallest degree of a cyclic isogeny between the \(i\)-th and \(j\)-th curve in the isogeny class, in the LMFDB numbering.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.